R E S E A R C H
Open Access
Coupled fixed points for multivalued
mappings in fuzzy metric spaces
Zheyong Qiu and Shihuang Hong
**Correspondence:
hongshh@hotmail.com Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China
Abstract
In this paper, we establish two coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered fuzzy metric spaces. The theorems presented extend some corresponding results due to ordinary metric spaces. An example is given to illustrate the usability of our results.
Keywords: coupled fixed point; multivalued contractive mapping; fuzzy metric space; partially ordered set
1 Introduction
In , Nadler [] extended the famous Banach contraction principle from single-valued mappings to multivalued mappings and proved the existence of fixed points for contractive multivalued mappings in complete metric spaces. Since then, the existence of fixed points for various multivalued contractive mappings has been studied by many authors under dif-ferent conditions. For details, we refer to [–] and the references therein. For instance, in [] Ćirić has proved a fixed point theorem for the single-valued mappings satisfying some contractive condition. Samet and Vetro [] extended this result to multivalued mappings and proved the existence of a coupled fixed point theorem for the multivalued contraction. One of the most important problems in fuzzy topology is to obtain an appropriate con-cept of fuzzy metric spaces. This problem has been investigated by many authors from dif-ferent points of view. In particular, George and Veeramani [, ] introduced and studied the notion of fuzzy metricMon a setXwith the help of continuoust-norms introduced in [], and from now on, when we talk about fuzzy metrics, we refer to this type. Fuzzy metric spaces have many applications. In particular, on the fuzzy metric space, by using some topological properties induced by this kind of fuzzy metrics, there are several fixed point results established. Some instances of these works are in [–]. In fact, fuzzy fixed point results are more versatile than the regular metric fixed point results. This is due to the flexibility which the fuzzy concept inherently possesses. For example, the Banach con-traction mapping principle has been extended in fuzzy metric spaces in two inequivalent ways in [, ]. Fuzzy fixed point theory has a developed literature and can be regarded as a subject in its own right (see []).
In recent times, the existence of common or coupled fixed points of a fuzzy version for multiple mappings has attracted much attention. We mention that the coupled fixed point results were proved by Sedghiet al.[], which is a fuzzy version of the result of []. Choudhury [] further extended the result of [] and provided the existence re-sults of coupled coincidence points for compatible mappings in partially ordered fuzzy
metric spaces. After that common coupled fixed point results in fuzzy metric spaces were established by Hu []. However, to the best of our knowledge, few papers were devoted to fixed point problems of multivalued mappings in fuzzy metric spaces (see [, ]).
The aim of this paper is to present two new coupled fixed point theorems for two multi-valued mappings in the fuzzy metric space. The idea of the present paper originates from the study of an analogous problem examined by Sametet al.[] in regular partially ordered metric spaces due to Ćirić []. Our results give a significant extension of some correspond-ing results. An example is also given to illustrate the suitability of our results.
2 Preliminaries
To set up our main results in the sequel, we recall some necessary definitions and prelim-inary concepts in this section.
Definition .[] The binary operation∗: [, ]×[, ]→[, ] is called a continuous t-norm if the following properties are satisfied:
(T) a∗ =afor alla∈[, ],
(T) a∗b≤c∗dwhenevera≤candb≤dfor eacha,b,c,d∈[, ], (T) ∗is continuous, associative and commutative.
In this sequel, we further assume that∗satisfies
(T) a∗b≥abfor alla,b∈[, ].
For examples oft-norm satisfying the conditions (T)-(T), we enumeratea∗b=ab, a∗b=min{a,b}anda∗b=ab/max{a,b,λ}for <λ< , respectively.
Definition .[] The triplet (X,M,∗) is said to be a fuzzy metric space ifXis an arbi-trary nonempty set,∗is a continuoust-norm andMis a fuzzy set onX×X×(, +∞) satisfying the following conditions for allx,y,z∈X,s,t> :
(F) M(x,y,t) > ,
(F) M(x,y,t) = for allt> if and only ifx=y, (F) M(x,y,t) =M(y,x,t),
(F) M(x,y,t)∗M(y,z,s)≤M(x,z,t+s)and (F) M(x,y,·) : (, +∞)→[, ]is continuous.
In this sense, (M,∗) is called a fuzzy metric onX.
Example .[] LetXbe the set of all real numbers anddbe the Euclidean metric. Let a∗b=min{a,b}for alla,b∈[, ]. For eacht> , x,y∈X, letM(x,y,t) = t+dt(x,y). Then (X,M,∗) is a fuzzy metric space.
Let (X,M,∗) be a fuzzy metric space. Fort> and <r< , the open ballB(x,t,r) with centerx∈Xis defined by
B(x,t,r) =y∈X:M(x,y,t) > –r.
such that{xn} topologically converges tox. In fact, the topological convergence of
se-quences can be indicated by the fuzzy metric as follows.
Definition .[] Let (X,M,∗) be a fuzzy metric space.
(i) A sequence{xn}inXis said to be convergent to a pointx∈Xif limn→∞M(xn,x,t) = for anyt> .
(ii) A sequence{xn}inXis called a Cauchy sequence iflimn→∞M(xn+p,xn,t) = for
anyt> and a positive integerp.
(iii) A fuzzy metric space(X,M,∗), in which every Cauchy sequence is convergent, is said to be complete.
ByCB(X) we denote the collection consisting of all nonempty closed subsets ofX. For C∈CB(X) andx∈X, we define
M∇(C,x,t) =M∇(x,C,t) =supM(z,x,t) :z∈C. ()
Lemma . If A∈CB(X),x∈A if and only if M∇(A,x,t) = for all t> .
Proof SinceM∇(A,x,t) =sup{M(x,y,t) :y∈A}= , there exists a sequence{yn} ⊂Asuch
thatM(x,yn,t) > –n. Letn→ ∞, we haveyn→x. FromA∈CB(X) it follows thatx∈A.
Conversely, ifx∈A, we haveM∇(A,x,t) =sup{M(x,y,t) :y∈A} ≥M(x,x,t) = for any t> . This implies thatM∇(A,x,t) = for allt> .
Definition . An element (x,y)∈X×Xis a coupled fixed point ofF:X×X→CB(X) if
x∈F(x,y) and y∈F(y,x).
At the end of this section, we introduce the following necessary notions.
The functionf :X×X×(,∞)→(,∞) is said to be uniformly upper semi-continuous with respect to t∈(,∞) on X×X if limn→∞(xn,yn) = (x,y) implies that f(x,y,t)≥ lim supn→∞f(xn,yn,t) for{xn},{yn} ⊂Xand allt∈(,∞).
LetXbe a nonempty set endowed with a partial orderand letG:X→Xbe a given mapping. We recall the set
=(x,y)∈X×X:GxGy
and the binary relationRonCB(X) defined by
ARB ⇔ A×B⊂ forA,B∈CB(X).
Definition . Let F :X ×X→CB(X) be a given mapping. We say that F is a -symmetric mapping if and only if
3 Main results
In order to prove our main results, we need the following hypothesis.
(H) The functionf:X×X×(,∞)→(, ]defined by
f(x,y,t) =M∇x,F(x,y),t+M∇y,F(y,x),t
withF:X×X→CB(X)is uniformly upper semi-continuous with respect to
t∈(,∞)onX×X.
Theorem . Let(X,M,∗)be a complete fuzzy metric space with the partial orderand (x,y)∈.Let F:X×X→CB(X)be a-symmetric mapping and satisfy that
(i) the condition(H)holds and
(ii) for any(x,y)∈,ifg(x,y) =inft>f(x,y,t) < ,then there existu∈F(x,y)and v∈F(y,x)with
ϕg(x,y)M(x,u,t) +M(y,v,t)≥f(x,y,t) ()
such that
f(u,v,t)≥ϕg(x,y)M(x,u,t) +M(y,v,t), ()
where,t> ,the functionϕ: [, ]→[, +∞)is nonincreasing,ϕ(r) > for≤r< andlimr→–ϕ(r) > .
Then F admits a coupled fixed point on X×X.
Proof Ifg(x,y) = , then it is easy to see that (x,y)∈X×Xis a coupled fixed point ofF. Otherwise,g(x,y) < and the definition ofϕguarantees thatϕ(g(x,y)) > for each (x,y)∈ X×X. By virtue of the definition ofM∇, there existu∈F(x,y) andv∈F(y,x) for any given (x,y)∈X×Xsuch that
ϕg(x,y)M(x,u,t)≥M∇x,F(x,y),t,
ϕg(x,y)M(y,v,t)≥M∇y,F(y,x),t.
This implies that, for each (x,y)∈X×X, there existu∈F(x,y) andv∈F(y,x) such that () holds. Thus, for arbitrarily given (x,y)∈, eitherg(x,y) = , then we get our desired result, org(x,y) < . In this case, by the condition (ii), we can choosex∈F(x,y) and y∈F(y,x) such that
ϕg(x,y)
M(x,x,t) +M(y,y,t)
≥f(x,y,t), ()
ϕg(x,y)
M(x,x,t) +M(y,y,t)
≤f(x,y,t). ()
From () and () it yields that
f(x,y,t)≥
ϕg(x,y)
ϕg(x,y)
M(x,x,t) +M(y,y,t)
≥ϕg(x,y)
Note thatF is a -symmetric mapping. By Definition . we haveF(x,y)RF(y,x), which further implies (x,y)∈. If g(x,y) = , then (x,y) is a coupled fixed point of F and our desired result comes out. Otherwise, g(x,y) < , which implies that ϕ(g(x,y)) > . Again, the condition (ii) guarantees that there exist x ∈F(x,y) and y∈F(y,x) such that
ϕg(x,y)
M(x,x,t) +M(y,y,t)
≥f(x,y,t),
ϕg(x,y)
M(x,x,t) +M(y,y,t)
≤f(x,y,t).
Hence, we obtain (x,y)∈and
f(x,y,t)≥
ϕg(x,y)
f(x,y,t).
Continuing this process, we can choose the sequences{xn},{yn} ⊂Xsuch that either
g(xn,yn) = , which implies that (xn,yn) is a coupled fixed point ofF, org(xn,yn) < , which
impliesϕ(g(xn,yn)) > . In this case, we have
(xn,yn)∈, xn+∈F(xn,yn), yn+∈F(yn,xn), ()
ϕg(xn,yn)
M(xn,xn+,t) +M(yn,yn+,t)
≥f(xn,yn,t) ()
forn= , , , . . . and
f(xn+,yn+,t)≥
ϕg(xn,yn)
f(xn,yn,t) forn= , , , . . . . ()
It follows that the sequence{f(xn,yn,t)}is increasing from () and upper bounded from
the definition off. Therefore, this deduces{f(xn,yn,t)}is convergent, say,
lim
n→∞f(xn,yn,t) =α(t).
We assertα(t)≡ fort> . Suppose that this is not true, then there existst> such that <α(t) < . Note thatg(xn,yn)≤f(xn,yn,t), we haveϕ(g(xn,yn))≥ϕ(f(xn,yn,t)) for eachn∈Nsinceϕis nonincreasing. By means of this, taking the limit on both sides of () witht=tand having in mind the assumptions ofϕ, we have
α(t)≥ lim
f(xn,yn,t)→α(t)
ϕg(xn,yn)
f(xn,yn,t)
≥ lim
f(xn,yn,t)→α(t)–
ϕf(xn,yn,t)
α(t) >α(t)
a contradiction. Thusα(t)≡ for allt> .
We are in a position to verify that both{xn}and{yn}are Cauchy sequences in (X,M,∗).
Let
β= lim f(xn,yn,t)→–
ϕg(xn,yn)
Note that {g(xn,yn)} is nondecreasing, we obtain that {ϕ(g(xn,yn))} is nonincreasing.
Hence,β exists. Sinceϕ(g(xn,yn)) > , the properties ofϕ guarantee thatβ> . Let the
real numberpwithβ>p> andp>βbe any fixed. For anyε> withε<min{,p–β}, there existsn∈Nsuch that
β+ε≥
ϕg(xn,yn)
for eachn≥n.
On the other hand, we easily see that there existsn∈Nsuch that
ϕg(xn,yn)
>p for eachn≥n.
Substituting this inequality in (), we have
f(xn+,yn+,t) >pf(xn,yn,t) for allt> andn≥n.
For anym∈N, we inductively obtain
f(xn+m,yn+m,t) >pmf(xn,yn,t) for allt> andn≥n. ()
Letn≥max{n,n}. By means of this, together with () and () withn=n, one has
(β+ε)M(xn+m,xn+m+,t) +M(yn+m,yn+m+,t)
>pmf(xn,yn,t).
By virtue ofα(t)≡, we havef(xn,yn,t)≥ –εwhennis large enough. In addition, the hypothesis ofεimplies thatpm≥β+εform≥. Consequently, we can choose that
n∈Nis large enough such that
M(xn,xn+,t) +M(yn,yn+,t) > –ε for anyt> andn≥n,
which implies that, by the arbitrariness ofε,
M(xn,xn+,t) > –ε, M(yn,yn+,t) > –ε for anyt> andn≥n. ()
In order to prove that {xn}and{yn}are Cauchy sequences, for any given > , it is
sufficient to check
M(xn,xm,t) > –, M(yn,ym,t) > – ()
for anym>n≥n+ andt> . Virtually, for anyn≥n+ , ifm=n+ , then () is valid by () withε≤. Inductively suppose that () is valid form=k>n, eacht> and n≥n+ . Then, in view of the hypotheses (F) and (T), we have
M(xn,xk+,t)≥M(xn,xk,t/)∗M(xk,xk+,t/)≥M(xn,xk,t/)M(xk,xk+,t/).
Now, the inductive assumption shows thatM(xn,xk,t/) > –and the first inequality in
() guaranteesM(xk,xk+,t/) > –ε. Hence we have
Letε→, we haveM(xn,xk+,t) > –. By induction, we get that the first inequality in () is true. The proof of the second inequality in () is analogous.
Finally, by means of the completeness of (X,M,∗), there existsz= (z,z)∈X×Xsuch that
lim
n→∞xn=z and n→∞lim yn=z.
In what follows, we prove that z is a coupled fixed point ofF. Since f is upper semi-continuous, fromlimn→∞f(xn,yn,t) =α(t)≡ we get
≥f(z) =M∇z,F(z,z),t
+M∇z,F(z,z),t
≥lim sup
n→∞ f(xn,yn,t) = .
This impliesM∇(z,F(z,z),t) +M∇(z,F(z,z),t) = ; moreover, we have
M∇z,F(z,z),t
= and M∇z,F(z,z),t
= .
From Lemma . it follows thatz∈F(z,z) andz∈F(z,z), that is,z= (z,z) is a
cou-pled fixed point ofF. The proof is completed.
Theorem . Let(X,M,∗)be a complete fuzzy metric space endowed with a partial or-der,=∅,and let F:X×X→CB(X)be a-symmetric multivalued mapping satisfying that
(i) the(H)holds and
(ii) for any(x,y)∈,there existu∈F(x,y)andv∈F(y,x)withh(x,y,u,v) < and
ϕh(x,y,u,v)M(x,u,t) +M(y,v,t)≥f(x,y,t) ()
such that
f(u,v,t)≥ϕh(x,y,u,v)M(x,u,t) +M(y,v,t), ()
whereh(x,y,u,v) =inft>{M(x,u,t) +M(t,v,t)}< forx,y,u,v∈X,the function ϕ: [, ]→[, +∞)is nonincreasing and satisfies thatϕ(r) > for≤r< and
lim infr→s–ϕ(r) > for eachs∈(, ). Then F admits a coupled fixed point on X×X.
Proof Ifh(x,y,u,v) = , then it is easy to see thatx=uandy=v. In this case, (x,y) is a coupled fixed point ofFifu∈F(x,y) andv∈F(y,x). Otherwise,h(x,y,u,v) < and the definition ofϕ guarantees thatϕ(h(x,y,u,v)) > for each (x,y)∈X×X. As an analogy of the argument of Theorem ., we can construct the sequences{xn},{yn} ⊂Xsuch that
eitherh(xn,yn,xn+,yn+) = , which implies that (xn,yn) is a coupled fixed point ofF, or
h(xn,yn,xn+,yn+) < , which impliesϕ(h(xn,yn,xn+,yn+)) > . In this case, we have
(xn,yn)∈, xn+∈F(xn,yn), yn+∈F(yn,xn), ()
ϕh(xn,yn,xn+,yn+)
M(xn,xn+,t) +M(yn,yn+,t)
forn= , , , . . . and
f(xn+,yn+,t)≥
ϕh(xn,yn,xn+,yn+)
f(xn,yn,t) forn= , , , . . . . ()
Again, () guarantees that the sequence{f(xn,yn,t)}is increasing and upper bounded.
Therefore, there existsα: (,∞)→(, ] such that
lim
n→∞f(xn,yn,t) =α(t) fort> .
On the other hand, note that the sequence{hn}withhn={h(xn,yn,xn+,yn+)}is bounded, we have
lim sup
n→∞ hn=η
for someη∈(, ]. By virtue of (), combining (), we have
M(xn,xn+,t) +M(yn,yn+,t)≤f(xn,yn,t)
for anyn∈Nandt> , which implies thathn≤f(xn,yn,t). Moreover, one hasη≤α(t) for
allt> .
In what follows, we prove thatη= . To this end, we first proveα(t)≡ for allt> . Sup-pose, to the contrary, that there existst> such thatα(t) < . The monotonicity of the sequence{f(xn,yn,t)}andhn≤f(xn,yn,t) yields thatη≤α(t) andϕ(hn) > for alln∈N.
In addition, there exists the subsequence of{hn},{hnk}, such thatlimk→∞hnk=η –< . In
view of the hypothesis ofϕ, we havelim infk→∞ ϕ(hnk) > . By means of this and the property ofϕ, taking the limit on both sides of the following inequality:
f(xnk+,ynk+,t)≥
ϕ(hnk)f(xnk,ynk,t),
we obtainα(t) =lim infk→∞f(xnk+,ynk+,t)≥lim infk→∞[ ϕ(hnk)f(xnk,ynk,t)] >α(t), a contradiction. Hence,α(t)≡. Ifη< , repeating the above proceeding, we can obtain the contradiction of > . Consequently,η= .
Finally, byη= , it is easy to see that the sequences{xnk}and{ynk}satisfy (). Following the lines of the arguments of Theorem ., we can obtain that{xnk}and{ynk}are Cauchy sequences. The rest of the proof is the same as that of Theorem .. This completes our
proof.
4 An example
In this section, we conclude the paper with the following example.
Example . LetX= [,∞),a∗b=abfor alla,b∈[, ] and
M(x,y,t) =
⎧ ⎨ ⎩
, x=y,
Then (X,M,∗) is a fuzzy metric space (see []). LetXbe endowed with the usual order≤. Define the functionG:X→XbyG(x) =cfor anyx∈X, wherecis a positive constant. Then= (,∞)×(,∞). Define the multivalued functionF:X×X→CB(X) by
F(x,y) =
⎧ ⎨ ⎩ [ √ y+ , √ x+y+
], ≤y≤,
{√y+
}, y> ,
for allx∈X.
Conclusion F admits a coupled fixed point in X×X.
Proof It is not hard to see thatXis complete andFis a-symmetric mapping. Moreover,
M∇x,F(x,y),t=
x(√y+ ), M
∇y,F(y,x),t=
(√x+ )y.
Adding the above two formulae together, we obtain
f(x,y,t) = x(√y+ )+
(√x+ )y for (x,y,t)∈X×X×(, +∞).
f is obviously uniformly upper semi-continuous with respect tot∈(, +∞),i.e., (H) is satisfied. Let the functionϕ: [, ]→[,∞) be defined by
ϕ(t) = – t
, t∈(, ].
For anyx,y∈X, we takeu=
√ y+ ,v=
√ x+
, thenu∈F(x,y),v∈F(y,x) and
M(x,u,t) =
x(√y+ ), M(y,v,t) = (√x+ )y.
Therefore,u,vsatisfy the inequality () and
f(u,v,t) = u(√v+ )+
(√u+ )v
=
(√y+ )
√ x+ +
+
(√x+ )
√ y+ +
=
x(√y+ )·
√x
√
x+ + √
+
y(√x+ )·
√y
√
y+ + √
≥
x(√y+ )+
(√x+ )y
≥ϕg(x,y)M(x,u,t) +M(y,v,t).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors read and approved the final manuscript.
Acknowledgements
The authors wish to express their hearty thanks to the anonymous referees for their valuable suggestions and comments. Supported by Natural Science Foundation of Zhejiang Province (LY12A01002).
Received: 11 January 2013 Accepted: 4 June 2013 Published: 25 June 2013 References
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